There is an implicit binary choice here, so “whether” fits. Both work, although I, for one, prefer to use “whether” for binary choices and “if” when there are more options. This is similar to my preference for “between” only for two things and “among” for more than two.

The phrase “I’m wondering if… can…” needs a noun or pronoun between “if” and “can”. As soon as you try to remove that (by moving it out to “The monkey who…”), the phrase stops being grammatical. We’d understand you, but it would require significant effort to parse the sentence. That seems to be what makes this sound strange, no matter what we try to do with it.

I don’t know whether other languages can do this, but English can’t.

• Omnibus
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This is why they never hold the Canadian championships anywhere near Saskatchewan.

Well.

Another reason.